Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
580 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C24 |
| Fitting | C3×C4×C7×Q8 | C2 |
| hypercenter | C4×Q8 | D21 |
| hypocenter | C3×C7 | #548 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66732 | C2 | semidirect |
| 2. C2×Q8×D21 | C2 | semidirect |
| 3. C3×C4×C7×Q8 | C2 | semidirect |
| 4. #10621380 | C2 | semidirect |
| 5. #10621378 | C2 | semidirect |
| 6. #10621358 | C2 | semidirect |
| 7. C4×D84 | C2 | semidirect |
| 25. C4×C7×Q8 | D3 | semidirect |
| 33. C3×C7×Q8 | D4 | semidirect |
| 60. C3×C4×Q8 | D7 | semidirect |
| 71. C2×D21 | #41 | semidirect |
| 80. C7×Q8 | D12 | semidirect |
| 115. C4×Q8 | D21 | semidirect |
| 131. C3×Q8 | D28 | semidirect |
| 142. C3×C7 | #548 | semidirect |
| 177. Q8 | D84 | semidirect |
| 187. C7 | #5576 | semidirect |
| 205. C3 | #90267 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66732 | 2 |
| C2×Q8×D21 | 2 |
| C3×C4×C7×Q8 | 2 |
| #10621380 | 2 |
| #10621378 | 2 |
| #10621358 | 2 |
| C4×D84 | 2 |
| #90267 | 3 |
| #5576 | 7 |